Why does Yuki love Yamato so much?
After she lost the chess game with the wager of her servitude, she could have just ignored him and move on with her life and duty. But losing to a human,a creation of god, made her question the reality of god himself. As a divine being meant to be above mortals, losing to one inevitably made that belief crumble and thus destroyed the principle she lived by since the beginning of time. Yamato being the anomaly that brought her clarity and with it a new purpose and object of devotion, from which her romantical and sexual attraction formed, made her put him on a pedestal even above god and as framed him as an untouchable concept. This obsession, at first just admiration, caused by her inability to predict and or understand him, developed into the romantical love she feels today only moments after, as all this happened within seconds.
What is her goal?
Yuki strives to make Yamato proud, make his every wish come true and preferably without him even having to ask. She takes pride in being his one and only, the best there is for him, without anyone else even comparing. Her final goal is a state of being where only the two exist or matter or both. In her mind, if she is the only thing that exists, what he will feel towards her will inevitably be love. Him expressing that he would like to feel and experience emotions plays into that, as it serves both as internal justification and genuine desire to help him.
How is her perception of Yamato overall and how does she act with him?
Yamato is without a doubt the center of her world, as her every thought and action revolves around him and him alone. She will distinguish between her own needs, his needs and the needs of outsiders. His needs come first, then hers, and others do not even make it into the equation. She will do so with tender care, shower him with love and attention because what she feels is so overwhelming that she cannot contain it all, which she also does not try to do. Although she views him as her god, she rarely views the things she does as religious, because she usually only sees the two of them and pictures it more as a romantic fantasy rather than acts of devotion done in servitude. But at the same time she views herself as his property and closest "folllwer", but only in comparison to other people. Her intimate behavior around him ties directly into the trust she has in him, as he could make up anything and she would support him no matter what, unless it includes other people, which will cause protests on her part. She will however frame it purposefully none threatening as she would never force him to do anything, much rather would she want to be forced to comply, as it gives her internal justification for whatever follows that displeases her. There is also no sequence of events that ends in her purposefully hurting him, even if it were to reach a higher goal centered around him, with the sole exception of him ordering her to, which she would still complain if she has the opportunity and time for.
What are other people to her?
The same way she takes pride in being Yamato's one, he takes pride in being above everyone else. She absolutely hates all life that is not Yamato, and is unable to form a meaningful connection with anything that is not him. Even if someone where to come and beat Yamato, proving their superiority over him, which theoretically would create the same scenario that happened with god and Yamato, this would not change anything. Yuki's love is not like her previous devotion to god, and is instead fundamentally unshakable. She has no empathy for people's suffering or the consequences of her actions affecting others, each time she seems to do she is playing a role in an attempt to either seduce Yamato, act within social norms or has a warped, even sadistic sense of sarcasm in her head with which she mocks them internally while pretending to care.
Patterns and why she acts on them:
A prominent pattern is her tendency to act extremely violent and I proportional to the "offence" she reacts on. Even though Yamato himself dislikes physical violence, he does not explicitly state disliking her doing it, which is a way she gains indirect approval through him each time she acts on her impulses. All her impulses stem from other people interacting with her, sometimes geniusely offensive, sometimes her own definitions of what is offensive or even just boredom and general disinterest cause her to spiral down into obscurity and generally degrading thoughts, which she only acts on once Yamato gives her permission or order, or if he is not part of the scene at all and also has no negative consequences from it, which she always calculates before acting out.
Another core aspect of her is her manipulative nature and behavior, which usually affects those around her, excluding Yamato, negatively. Operation-ByeBye is a great example of everything mentioned above, which is why the chapter also included the title "Strings of Yuki". In general, she acts manipulative all the time, even towards Yamato. Though he usually calls her out, but not always, he does notice it every time. The only time he ever explicitly seemed to mind, was when she made him kill Satoshi. This ended up being one of Yuki's worst regrets and she was plagued with guilt even months after, which she did not show and even Yamato did not notice. She generally does not directly enjoy any of her manipulative or violent actions, but does so in the end because it is framed as helping Yamato, or ensuing their love and togetherness, which often blends in with the slight sadism mentioned earlier. The fact that none of these actions bother her to see or do herself or make orhers do them for her also plays role in this.
The final pattern I will talk about is her possessive, over protective thought process. One of her core motivations is jealousy and compulsive need for his constant attention. Even though Yamato compares it to a child multiple times, it is only accurate out of his perspective as he views most emotions as "childish", but in this case Yuki's actions are calculated and methodical, which to him still seems childish, but in to "normal" people would change many things. Once she spirals down a protective thought, it almost always ends or includes violence towards the target. Said target is an intruder into the safe space created by her for Yamato and her alone. When she refers to him being stolen, she usually means his attention, which she needs everything of, not her being replaced. Her confidence is both grounded in her usefulness but at the same time originates from her superiority complex where she is literally irreplaceable. Once her violent thoughts start, it is usually up to Yamato both directly and indirectly if she acts on them. As example, should he completely ignore the "intruder", would her thoughts "just" be dehumanizing, degrading and generally hurtful, but most likely stay just thoughts. However that requires her to think that the existence of said person does not affect Yamato's and her life at all, despite them trying to. Should this not be the case, begins her defense mechanism which usually involves her switching up demeanor completely or just internally. She will immediately have constructed a plan of the scale comparable to Operation-ByeBye or larger, and will act with the intent to kill or torture, though the latter she has trouble to amplify in a school environment without causing trouble for Yamato, which she will try to avoid at all costs and remains her main priority.